A student finds several pairs of triangular numbers that average to a third one, and so
wonders how many more such triples exist — and how to generate
them. With a few inspired variable substitutions and some modular arithmetic, Doctor
Jacques responds, then suggests a few new questions to explore.

If a rational number can be found between any two irrationals, and the set of
irrationals are uncountably infinite, does that mean that the rationals are also
uncountable? Doctor Peterson points up the flaw in a student's assumption about what
to conclude from a failed mapping.

I've used a computer to evaluate [2 + sqrt(3)]^50 and the answer is
extremely close to being an integer. I've tried various expansions of
the expression to try and determine why it's so close to an integer,
but haven't gotten anywhere. Do you have any idea why?